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Subdivision, interpolation and splines

Goosen, Karin Michelle (2000-03)

Thesis (MSc)--University of Stellenbosch, 2000.

Thesis

ENGLISH ABSTRACT: In this thesis we study the underlying mathematical principles of stationary subdivision,
which can be regarded as an iterative recursion scheme for the generation of smooth curves
and surfaces in computer graphics. An important tool for our work is Fourier analysis, from
which we state some standard results, and give the proof of one non-standard result. Next,
since cardinal spline functions have strong links with subdivision, we devote a chapter to this
subject, proving also that the cardinal B-splines are refinable, and that the corresponding
Euler-Frobenius polynomial has a certain zero structure which has important implications
in our eventual applications. The concepts of a stationary subdivision scheme and its convergence
are then introduced, with as motivating example the de Rahm-Chaikin algorithm.
Standard results on convergence and regularity for the case of positive masks are quoted and
graphically illustrated.
Next, we introduce the concept of interpolatory stationary subdivision, in which case
the limit curve contains all the original control points. We prove a certain set of sufficient
conditions on the mask for convergence, at the same time also proving the existence and
other salient properties of the associated refinable function. Next, we show how the analysis
of a certain Bezout identity leads to the characterisation of a class of symmetric masks which
satisfy the abovementioned sufficient conditions. Finally, we show that specific special cases
of the Bezout identity yield convergent interpolatory symmetric subdivision schemes which
are identical to choosing the corresponding mask coefficients equal to certain point evaluations
of, respectively, a fundamental Lagrange interpolation polynomial and a fundamental
cardinal spline interpolant. The latter procedure, which is known as the Deslauriers-Dubuc
subdivision scheme in the case of a polynomial interpolant, has received attention in recent
work, and our approach provides a convergence result for such schemes in a more general
framework.
Throughout the thesis, numerical illustrations of our results are provided by means of
graphs.